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Properties

Label 2-2016-504.331-c0-0-1
Degree 22
Conductor 20162016
Sign 0.592−0.805i0.592 - 0.805i
Analytic cond. 1.006111.00611
Root an. cond. 1.003051.00305
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + i·5-s + (0.866 − 0.5i)7-s + (−0.499 − 0.866i)9-s + 11-s + (0.866 − 0.5i)13-s + (−0.866 − 0.5i)15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + 0.999i·21-s − i·23-s + 0.999·27-s + (−0.866 − 0.5i)29-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)35-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + i·5-s + (0.866 − 0.5i)7-s + (−0.499 − 0.866i)9-s + 11-s + (0.866 − 0.5i)13-s + (−0.866 − 0.5i)15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + 0.999i·21-s − i·23-s + 0.999·27-s + (−0.866 − 0.5i)29-s + (−0.5 + 0.866i)33-s + (0.5 + 0.866i)35-s + ⋯

Functional equation

Λ(s)=(2016s/2 ΓC(s) L(s)=((0.592−0.805i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2016s/2 ΓC(s) L(s)=((0.592−0.805i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 20162016    =    25⋅32⋅72^{5} \cdot 3^{2} \cdot 7
Sign: 0.592−0.805i0.592 - 0.805i
Analytic conductor: 1.006111.00611
Root analytic conductor: 1.003051.00305
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2016(79,⋅)\chi_{2016} (79, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2016, ( :0), 0.592−0.805i)(2,\ 2016,\ (\ :0),\ 0.592 - 0.805i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.1948756501.194875650
L(12)L(\frac12) ≈\approx 1.1948756501.194875650
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
7 1+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
good5 1−iT−T2 1 - iT - T^{2}
11 1−T+T2 1 - T + T^{2}
13 1+(−0.866+0.5i)T+(0.5−0.866i)T2 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2}
17 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
19 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
23 1+iT−T2 1 + iT - T^{2}
29 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
31 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
37 1+(0.866+0.5i)T+(0.5+0.866i)T2 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2}
41 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
43 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
59 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
79 1+(1.73−i)T+(0.5−0.866i)T2 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2}
83 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
89 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
97 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.587589548471870389599470092871, −8.730531829315067923124603784024, −7.955477714235190748248262014195, −6.92870572666423559025515893823, −6.29794826676785517414818559118, −5.48848765485006413602720253093, −4.45122840173151208539054881816, −3.78374022559574872057664383171, −2.92414905440633866336208172549, −1.28439242315768460532123388816, 1.29218108148143774963332762647, 1.74203795383929641651888265714, 3.39864777543165075511724832125, 4.54313980455455922116515121411, 5.41969378321553787935723681215, 5.83076104638045905265364285067, 6.97672274278337315943678076104, 7.61242454489178719072553868603, 8.589198797342687419657004015702, 8.888943587485255001387938938330

Graph of the ZZ-function along the critical line